Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417692 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1098 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,9,11)(7,10,8,12)(13,14,18,22,20,15,19,23)(16,17,26,25,21,24,27,28);(5,7,9,8)(6,10,11,12)(13,15)(14,20)(16,25)(17,26)(18,22)(19,23)(21,28)(24,27);(6,11)(10,12)(13,16)(14,17)(15,24)(18,26)(19,27)(20,21)(22,25)(23,28);(2,4)(5,8,9,7)(6,12,11,10)(13,17,20,24)(14,21,15,16)(18,28,19,25)(22,26,23,27)Polycyclic presentation:
2^9|a^2=ef,b^2=f,d^2=fi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,e^d=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 337126);